76.21.13 problem 14 (a)

Internal problem ID [17777]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 14 (a)
Date solved : Tuesday, January 28, 2025 at 11:02:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{2}+y&=\delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 14.425 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)+1/2*diff(y(t),t)+y(t)=Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {4 \,{\mathrm e}^{\frac {1}{4}-\frac {t}{4}} \sqrt {15}\, \operatorname {Heaviside}\left (t -1\right ) \sin \left (\frac {\sqrt {15}\, \left (t -1\right )}{4}\right )}{15} \]

✓ Solution by Mathematica

Time used: 0.085 (sec). Leaf size: 40

DSolve[{D[y[t],{t,2}]+1/2*D[y[t],t]+y[t]==DiracDelta[t-1],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {4 e^{\frac {1}{4}-\frac {t}{4}} \theta (t-1) \sin \left (\frac {1}{4} \sqrt {15} (t-1)\right )}{\sqrt {15}} \]